Finite Group Databasebeta

← #8527

Subgroups — #8527

Order 256.

328 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#541C211yesmaximal, characteristic
2#1558C212yesmaximal, complemented
3#1534C212yesmaximal, complemented
4#1464C212yesmaximal
5#541—24no—
6#541—24no—
7C2×C4⋊Q8C2211yescharacteristic
8C2×#126—22no—
9C2×#125—22no—
10C2×#125C2211yescharacteristic
11C2×C22⋊Q8C412yes—
12C2×C4×Q8C412yes—
13C2×#117C2211yescharacteristic
14#432C2214yes—
15#333—24no—
16#330—24no—
17#324—22no—
18#323—22no—
19C22×Q8C2×C412yes—
20C23×C4C2×C411yescharacteristic
21C4⋊Q8—22no—
22C4⋊Q8D414yes—
23#126—216no—
24#125—216no—
25#125—22no—
26#125D414yes—
27C22⋊Q8—24no—
28C22⋊Q8—24no—
29C4×Q8—24no—
30C4×Q8—24no—
31#117—22no—
32#117—22no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4C2×C412yes—
38C2×C4⋊C4D412yes—
39C2×C22⋊C4—22no—
40C2×C22⋊C4D412yes—
41C2×C42—22no—
42C2×C42C2×C411yescharacteristic
43C2×C42C2311yescharacteristic
44C4⋊C8—24no—
45C22⋊C8—24no—
46#97—24no—
47C24C22⋊C411yescharacteristic
48C2×Q8—22no—
49C2×Q8—44no—
50C2×Q8C22⋊C414yes—
51C22×C4—22no—
52C22×C4—22no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—42no—
56C22×C4C22⋊C411yescharacteristic
57C22×C4C22⋊C412yes—
58C22×C4C22×C411yescharacteristic
59C22×C4C2×D412yes—
60C2×C8—44no—
61C4⋊C4—28no—
62C4⋊C4—28no—
63C4⋊C4—22no—
64C4⋊C4—28no—
65C4⋊C4—28no—
66C4⋊C4—24no—
67C4⋊C4C22⋊C414yes—
68C22⋊C4—28no—
69C22⋊C4—24no—
70C42—28no—
71C42—21no—
72C42—21no—
73C42C2×D412yes—
74C42C2×D412yes—
75C23—21no—
76C23—22no—
77C23—42no—
78C23C2×C22⋊C411yescharacteristic
79Q8—44no—
80Q8—44no—
81C2×C4—22no—
82C2×C4—22no—
83C2×C4—24no—
84C2×C4—24no—
85C2×C4—21no—
86C2×C4—22no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4—44no—
91C2×C4—44no—
92C2×C4—42no—
93C2×C4C2×C22⋊C411yescharacteristic
94C2×C4C2×C22⋊C411yescharacteristic
95C2×C4#12014yes—
96C8—84no—
97C22—22no—
98C22—42no—
99C22—41no—
100C22—82no—
101C22#37911yescharacteristic
102C22#41312yes—
103C4—24no—
104C4—21no—
105C4—21no—
106C4—42no—
107C4—44no—
108C4—41no—
109C4—44no—
110C2—22no—
111C2—81no—
112C2#178511yescharacteristic
113C2#179612yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #1558C2semidirect
3. #1534C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#5412
#15582
#15342
#14642